The solar energetic particles (SEPs) of consequence to space weather are accelerated at shock waves driven out from the Sun by fast coronal mass ejections (CMEs). In the large events, these great shocks fill half of the heliosphere. SEP intensity profiles change appearance with longitude. Events with significant intensities of greater than ten MeV protons occur at an average rate of approx. 13 per year near solar maximum and several events with high intensities of > 100 McV protons occur each decade. As particles stream out along magnetic field lines from a shock near the Sun, they generate waves that scatter subsequent particles. At high intensities, wave growth throttles the flow below the 'streaming limit.' However, if the shock maintains its strength, particle intensities can rise above this limit to a peak when the shock itself passes over the observer creating a 'delayed' radiation hazard, even for protons with energies up to approx. one GeV. The streaming limit makes us blind to the intensities at the oncoming shock, however, heavier elements such as He, O, and Fe probe the shape of the wave spectrum, and variation in abundances of these elements allow us to evade the limit and probe conditions at the shock, with the aid of detailed modeling. At high energies, spectra steepen to form a spectral 'knee'. The location of the proton spectral knee can vary from approx. ten MeV to approx. one GeV, depending on shock conditions, greatly affecting the radiation hazard. Hard spectra are a serious threat to astronauts, placing challenging requirements for shielding, especially on long-duration missions to the moon or Mars.